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In this paper, we deal with the second-order Hamiltonian system (∗)u¨−L(t)u+∇W(t,u)=0.$$ (*)\quad\ddot{u}-L(t)u+\nabla W(t, u)=0. $$ We establish some criteria which guarantee that the above system has at least one or infinitely many homoclinic solutions under the assumption that W(t,x)$W(t, x)$ is subquadratic at infinity and L(t)$L(t)$ is a real symmetric matrix and satisfies lim inf|t|→+∞[|t|ν−2inf|x|=1(L(t)x,x)]>0$$\liminf_{|t|\to+\infty} \Bigl[|t|^{\nu-2}\inf _{|x|=1}\bigl(L(t)x, x\bigr) \Bigr] >0 $$ for some constant ν<2$\nu<2$. In particular, L(t)$L(t)$ and W(t,x)$W(t, x)$ are allowed to be sign-changing.

New conditions on homoclinic solutions for a subquadratic second order Hamiltonian system